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In 1905, one of Einstein’s achievements was to establish the theory of Special Relativity from 2 single postulates and correctly deduce their physical consequences (some of them time later). The essence of Special Relativity, as we have seen, is that all the inertial observers must agree on the speed of light “in vacuum”, and that the physical laws (those from Mechanics and Electromagnetism) are the same for all of them. Different observers will measure (and then they see) different wavelengths and frequencies, but the product of wavelength with the frequency is the same. The wavelength and frequency are thus Lorentz covariant, meaning that they change for different observers according some fixed mathematical prescription depending on its tensorial character (scalar, vector, tensor,…) respect to Lorentz transformations. The speed of light is Lorentz invariant.

By the other hand, Newton’s law of gravity describes the motion of planets and terrrestrial bodies. It is all that we need in contemporary rocket ships unless those devices also carry atomic clocks or other tools of exceptional accuracy. Here is Newton’s law in potential form:

In the special relativity framework, this equation has a terrible problem: if there is a change in the mass density , then it must propagate everywhere instantaneously. If you believe in the Special Relativity rules and in the speed of light invariance, it is impossible. Therefore, “Houston, we have a problem”.

Einstein was aware of it and he tried to solve this inconsistency. The final solution took him ten years .

The apparent silly and easy problem is to develop and describe all physics in the the same way irrespectively one is accelerating or not. However, it is not easy or silly at all. It requires deep physical insight and a high-end mathematical language. Indeed, what is the most difficult part are the details of Riemann geometry and tensor calculus on manifolds. Einstein got private aid from a friend called Marcel Grossmann. In fact, Einstein knew that SR was not compatible with Newton’s law of gravity. He (re)discovered the equivalence principle, stated by Galileo himself much before than him, but he interpreted deeper and seeked the proper language to incorporante that principle in such a way it were compatible (at least locally) with special relativity! His “journey” from 1907 to 1915 was a hard job and a continuous struggle with tensorial methods…

Today, we are going to derive the Einstein field equations for gravity, a set of equations for the “metric field” . Hilbert in fact arrived at Einstein’s field equations with the use of the variational method we are going to use here, but Einstein’s methods were more physical and based on physical intuitions. They are in fact “complementary” approaches. I urge you to read “The meaning of Relativity” by A.Einstein in order to read a summary of his discoveries.

We now proceed to derive Einstein’s Field Equations (EFE) for General Relativity (more properly, a relativistic theory of gravity):

Be aware of the square root of the determinant of the metric as part of the volume element. It is important since the volume element has to be invariant in curved spacetime (i.e.,in the presence of a metric). It also plays a critical role in the derivation.

Step 2. We perform the variational variation with respect to the metric field :

Step 3. Extract out the square root of the metric as a common factor and use the product rule on the term with the Ricci scalar R:

Step 4. Use the definition of a Ricci scalar as a contraction of the Ricci tensor to calculate the first term:

A total derivative does not make a contribution to the variation of the action principle, so can be neglected to find the extremal point. Indeed, this is the Stokes theorem in action. To show that the variation in the Ricci tensor is a total derivative, in case you don’t believe this fact, we can proceed as follows:

Check 1. Write the Riemann curvature tensor:

Note the striking resemblance with the non-abelian YM field strength curvature two-form

.

There are many terms with indices in the Riemann tensor calculation, but we can simplify stuff.

Check 2. We have to calculate the variation of the Riemann curvature tensor with respect to the metric tensor:

One cannot calculate the covariant derivative of a connection since it does not transform like a tensor. However, the difference of two connections does transform like a tensor.

Check 3. Calculate the covariant derivative of the variation of the connection:

Check 4. Rewrite the variation of the Riemann curvature tensor as the difference of two covariant derivatives of the variation of the connection written in Check 3, that is, substract the previous two terms in check 3.

Check 5. Contract the result of Check 4.

Check 6. Contract the result of Check 5:

Therefore, we have

Q.E.D.

Step 5. The variation of the second term in the action is the next step. Transform the coordinate system to one where the metric is diagonal and use the product rule:

The reason of the last equalities is that , and then its variation is

Thus, multiplication by the inverse metric produces

that is,

By the other hand, using the theorem for the derivation of a determinant we get that:

since

because of the classical identity

Indeed

and moreover

so

Q.E.D.

Step 6. Define the stress energy-momentum tensor as the third term in the action (that coming from the matter lagrangian):

or equivalently

Step 7. The extremal principle. The variation of the Hilbert action will be an extremum when the integrand is equal to zero:

i.e.,

Usually this is recasted and simplified using the Einstein’s tensor

as

This deduction has been mathematical. But there is a deep physical picture behind it. Moreover, there are a huge number of physics issues one could go into. For instance, these equations bind to particles with integral spin which is good for bosons, but there are matter fermions that also participate in gravity coupling to it. Gravity is universal. To include those fermion fields, one can consider the metric and the connection to be independent of each other. That is the so-called Palatini approach.

Final remark: you can add to the EFE above a “constant” times the metric tensor, since its “covariant derivative” vanishes. This constant is the cosmological constant (a.k.a. dark energy in conteporary physics). The, the most general form of EFE is:

Einstein’s additional term was added in order to make the Universe “static”. After Hubble’s discovery of the expansion of the Universe, Einstein blamed himself about the introduction of such a term, since it avoided to predict the expanding Universe. However, perhaps irocanilly, in 1998 we discovered that the Universe was accelerating instead of being decelerating due to gravity, and the most simple way to understand that phenomenon is with a positive cosmological constant domining the current era in the Universe. Fascinating, and more and more due to the WMAP/Planck data. The cosmological constant/dark energy and the dark matter we seem to “observe” can not be explained with the fields of the Standard Model, and therefore…They hint to new physics. The character of this new physics is challenging, and much work is being done in order to find some particle of model in which dark matter and dark energy fit. However, it is not easy at all!

Dirichlet eta function

This function is indeed the Riemann zeta function with alternating plus/minus signs. In other words:

Applications: physmatics.

Related ideas: Riemann zeta function.

Reciprocal Riemann zeta function

Reciprocal zeta function is the following modification of the Riemann zeta function:

where the Möbius function is defined as follows

A number is said to be square-free if it is not divisible by a number which is a perfect square (excepting the number one). An alternative definition of the Möbius function is given by:

and where is the number of different primes dividing the number and is the number of prime factors of , counted with multiplicities. Clearly, the inequality is satisfied. Moreover, note that and is undefined.

Indeed, we also have:

This result is important for the so-called Dirichlet generating series:

By the other hand, since

taking the ratio between these last two results, we obtain the beautiful equation

The Liouville function is defined similarly to the Möbius function. If is a positive integer, it is:

Using the sum of the geometric series, we get:

while if we use the Liouville function, we could write

There is other remarkable family of infinite products

where again counts the number of distinct prime factors of and is the number of square-free divisors. Furthermore, if is a Dirichlet character of conductor N, so that is totally multiplicative and only depends on , and if is not coprime to N, then the following identity holds

Here it is convenient and common to omit the primes dividing the conductor from the product.

Hurwitz zeta function

It is the the generalization of Riemann zeta function given by the next sum:

Remark: the mathematica code for this function is Zeta[s,Q].

Multiple zeta value/Euler sum/Polyzeta

Multiple zeta values, also called polyzeta function or Euler sums are certain “coloured” generalizations (in several variables) of the Riemann zeta function:

Polylogarithm/Coloured polylogarithm

The polygogarithm is the following generalization of Riemann zeta function:

There are coloured versions of the polylogarithm:

Lerch-zeta function/Lerch-trascendent

The Lerch-zeta function is defined with the sum:

The Lerch trascendent is the function

Lerch-zeta function and Lerch trascendent are related through the functional equation

Mordell-Tornheim zeta values

Defined by Matsumoto in 2003, these zeta functions are:

Barnes zeta function

This function is the sum

where are numbers such that , and the sum is defined for all complex number s whenever .

Airy zeta function

Let be the zeros of the Airy function . Then, the Airy zeta function is the sum:

Arithmetic zeta function

The arithmetic zeta function over some scheme is defined to be the sum:

where the product is taken on every closed point of the scheme X.

The generalized Riemann hypothesis over the scheme is the hypothesis that the zeros of such arithmetic function, i.e, the feynmanity , and its poles are found in the next way:

inside the critical strip.

Artin-Mazur zeta function

Let us define:

1st. is the the set of fixed points of the nth iterated function of f.

2nd. is the cardinality of the set , i.e., the number of elements of such a set.

Then, the Artin-Mazur zeta function is the zeta function given by the next formula:

Dedekind zeta function

Let us define:

1st. is an algebraic number field.

2nd. is the range of non zero ideals of the ring of integers of K.

3rd. is the aboslute norm of I. When we get the usual Riemann zeta function.

Then, the Dedekind zeta function is the sum

where .

Epstein zeta function/Eisenstein series

where we have defined as the quadratic form . A related concept is the Eisenstein (not confuse with Einstein, please)

where and the sum is taken on every pari of coprime integers. Two integers A and B are said to be coprime (also spelled co-prime) or relatively prime if the only positive integer that evenly divides both of them is 1.

There is a relation with modular forms/automorphic forms as well. Let be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series of weight , where is an integer, by the series:

It is absolutely convergent to a holomorphic function of in the upper half-plane and its Fourier expansion given below shows that it can be extended to a holomorphic function at . It is a remarkable and surprising fact that the Eisenstein series is a modular form. Indeed, the key property is its -invariance. Explicitly if and then the next group property is satisfied

and is therefore a modular form of weight .

Remark: it is important to assume that otherwise it would be illegitimate to change the order of summation, and the -invariance would not remain. In fact, there are no nontrivial modular forms of weight 2. Nevertheless, an analogue of the holomorphic Eisenstein series can be defined even for although it would only be what mathematicians call a quasimodular form.

Ihara zeta function

This zeta function appears in graph theory and it has an amazing set of useful identities. The Ihara zeta function is the sum:

where the product runs over every prime walk p of the graph , i.e., it is taken over closed cycles such as with and is equal to the length of the cycle p.

The Ihara formula is a key result in graph theory

and there is the circuit rank, i.e., it is the cyclomatic number of an undirected graph G or the minimum number of edges necessary to remove from G all its cycles, making it into a forest (graph without cycles, a fores is only a disjoint union of “trees”). Finally, if is the Hashimoto’s edge adjacency operator, then

Matsumoto zeta function

A class of zeta functions defined by Matsumoto around 1990. They are functions

where is a prime number and is certain polynomial.

Minakshisundaram-Pleijel zeta function

A type of zeta function encoding the eigenvalues of a Lapalacian of a compact riemannian manifold . If and the eigenvalues of the Laplace-Beltrami operator are the set , then the Minakshisundaram-Pleijel zeta function is defined as the following series (where we removed the zero eigenvalues from the sum and , i.e., the real part of s is large enough):

Prime Zeta function

The next function was defined by Fröberg, Cohen and Glaisher, with the only subtle point of being careful to consider as a prime in the sum or not and the notation they used:

Note that such a function is a “prime” version of the Riemann zeta function:

Remark: Cohen used a different notation for . He used instead of the Fröberg’s and Glaisher notation.

Remark (II): Interestingly, the prime zeta function has the following behaviour close to the axis

where

This prime zeta function is related to the Riemann zeta function:

so

This equation and definition can be inverted (the original inversion procedure was carried by Glaisher around 1891, it is recalled by Fröberg about 1968, and it was studied later by Cohen, circa 2000):

Remark: the mathematica code for the prime zeta function is PrimeZetaP[s] and Zeta[s] for the Riemann zeta function.

Remark (II):

Remark (III): Fröberg (1968) stated that very little is known about the prime zeta function zeroes in the complex plane, i.e., the solutions to . Unlike the Riemann zeroes, it seems that prime zeta function zeroes are not on a straight line, but there is no known pattern, if any.

Remark (IV): Despite the divergence of , dropping the initial term and adding the Euler-Mascheroni constant provides a new constant! It is called Mertens constant. That is,

Remark (V): The Artins constant is related to as well

and where is the n-th Lucas number.

Remark (VI): The prime zeta function has the next asymptotical behaviour close to

Ruelle zeta function

Let’s define the following concepts:

1st. is certain function or map on a manifold M.

2nd. is the set of fixed points of the nth iterated function of f, being such an iterated function a finite value.

3rd. is certain function on M with values or entries in complex matrices. The case corresponds to the Artin-Mazur zeta function.

The Ruelle zeta function is the object defined with the series

Selberg zeta function

This zeta function is related to a compact ( of finite volume) Riemannian manifold. Assuming that certain manifolf M has constant curvature , it can be realized as a quotient of the Poincaré upper half plane

The Poincaré arc length is defined in this space as

and it can be shown to be invariant under fractional linear transformations

with and . Indeed, it is not hard to prove that the geodesics (curves minimizing the Poincaré arc length) are half lines and semicircles in H orthogonal to the real axis. Calling these lines as geodesics creates a model of hyperbolic geometry, i.e., a non-euclidean model for geometry where the 5th Euclid postulate is not longer valid. In fact, there are infinitely many geodesics through a fixed point not meeting a given geodesic. The fundamental group of M acts as a discrete group of transformations preserving distances between points. The favourite group between number theorists is called the modular group of matrices of determinant one and integer entries in the quotien space . However, the Riemann surface is noncompact, although it does have finite volume. Selberg introduced “prime numbers” in the compact surface to be “primitive cycles” or more precisely “primitive closed geodesics” C in M. There, the word “primitive” means that you can only go around the curve once. Furthermore, the Selberg zeta function, for large enough, is defined to be the sum

and where the product is extended over every primitive closed geodesics C in of Poincaré length . By the Selberg trace formula (which we are not goint to discuss here today), there is a duality between the lengths of the primes and the spectrum of the Laplace operator on M. Here, the Laplacian on M is

Indeed, it shows that one can show that the Riemann hypothesis (suitably modified to fit the situation) can be proved for Selberg zeta functions of compact Riemann surfaces! The closed geodesics in correspond to geodesics in H itself. One can show that the endpoints of such geodesics in the real line (note that the real line is the boundary of the set H) are fixed by hyperbolic elements of . That is, they are matrices

with trace . Primitive closed geodesics correspond to hyperbolic elements that generate their own centralizer in .

Shimizu zeta function

We define:

1st. , a totally algebraid number field.

2nd. , certain lattice in the field K.

3rd. , the subgroup of maximal rank of the group of the totally positive units preserving the lattice structure.

Then, the Shimizu zeta function arises in the form

Shintani zeta function

It is a generalized zeta series with the following formal definition

where are inhomogeneous functions of . Special cases of Shintani zeta function (or Shintani L-series, as they are also called by the mathematicians) are the Barnes zeta function or the Riemann zeta function.

Witten zeta function

Let G be a semisimple Lie group. The Witten L-series or Witten zeta function is defined by

This sum is taken over the equivalence classes of irreducible representations R of G. Considering a root system of rank equal to and with positive roots in , being all simple without loss of generality, the simple roots allow us to define the Witten zeta function as a function of several variables:

Zeta function of an operator

The zeta function of any (pseudo)-differential operator , or more generally any operator, can be defined as the following functional series:

and where the trace is taken over the values s where such number exists (i.e., the zero modes are removed). In fact, the zeta function of an arbitrary operator, that we can call the zetor, is the formal series:

It allow us to define the generalization of the determinant to -dimensional operators in the following non-trivial way:

Dirichlet L-function/L-series

They are the formal series

where is a Dirichlet character with conductor f, i.e.,

There, the generalized Bernoulli numbers are related to the L-series through the generating function above, and they satisfy the identity

p-adic zeta function

The p-adic analogue of the zeta function is defined with the following equation:

Moreover, we also define the zeta function at the infinite real prime:

The p-adic zeta function and the “real” prime zeta function (zeta function in the so-called “infinite prime”) satisfy the important adelic identity:

where , and is the classical Riemann zeta function. This adelic identity is just a special case of the adelic-type identity:

I was a mere undergraduate in the early years of the internet in my country when I began to read his TWF. If you have never done it, I urge to do it. Read him. He is a wonderful teacher and an excellent lecturer. John is now worried about global warming and related stuff, but he keeps his mathematical interests and pedagogical gifts untouched. I miss some topics about he used to discuss often before in his hew blog, but his insights about virtually everything he is involved into are really impressive. He also manages to share his entusiastic vision of Mathematics and Science. From pure mathematics to physics. He is a great blogger and scientist!

2nd. The professor Francis Villatoro. I am really grateful to him. He tries to divulge Science in Spain with his excellent blog ( written in Spanish language)

He is a very active person in the world of Spanish Science (and its divulgation). In his blog, he also tries to explain to the general public the latest news on HEP and other topics related with other branches of Physics, Mathematics or general Science. It is not an easy task! Some months ago, after some time reading and following his blog (as I do now yet, like with Baez’s stuff), I realized that I could not remain as a passive and simple reader or spectator in the web, so I wrote him and I asked him some questions about his experience with blogging and for advice. His comments and remarks were incredibly useful for me, specially during my first logs. I have followed several blogs the last years (like those by Baez or Villatoro), and I had no idea about what kind of style/scheme I should addopt here. I had only some fuzzy ideas about what to do, what to write and, of course, I had no idea if I could explain stuff in a simple way while keeping the physical intuition and the mathematical background I wanted to include. His early criticism was very helpful, so this post is a tribute for him as well. After all, he suggested me the topic of this post! I encourage you to read him and his blog (as long as you know Spanish or you can use a good translator).

Finally, let me express and show my deepest gratitude to John and Francis. Two great and extraordinary people and professionals in their respective fields who inspired (and yet they do) me in spirit and insight in my early and difficult steps of writing this blog. I am just convinced that Science is made of little, ordinary and small contributions like mine, and not only the greatest contributions like those making John and Francis to the whole world. I wish they continue making their contributions in the future for many, many years yet to come.

Now, let me answer the question Francis asked me to explain here with further details. My special post/log-entry number 50…It will be devoted to tell you why this blog is called The Spectrum of Riemannium, and what is behind the greatest unsolved problem in Number Theory, Mathematics and likely Physics/Physmatics as well…Enjoy it!

The Riemann zeta function is a device/object/function related to prime numbers.

In general, it is a function of complex variable defined by the next equation:

or

Generally speaking, the Riemann zeta function extended by analytical continuation to the whole complex plane is “more” than the classical Riemann zeta function that Euler found much before the work of Riemann in the XIX century. The Riemann zeta function for real and entire positive values is a very well known (and admired) series by the mathematicians. due to the divergence of the harmonic series. Zeta values at even positive numbers are related to the Bernoulli numbers, and it is still lacking an analytic expression for the zeta values at odd positive numbers.

The Riemann zeta function over the whole complex plane satisfy the following functional equation:

Equivalently, it can be also written in a very simple way:

where we have defined

Riemann zeta values are an example of beautiful Mathematics. From , then we have:

1) .

2) . The harmonic series is divergent.

3) . The famous Euler result.

4) . And odd zeta value called Apery’s constant that we do not know yet how to express in terms of irrational numbers.

5) .

6) . Trivial zeroes of zeta.

7) , where are the Bernoulli numbers. The first 13 Bernoulli numbers are:

8) We note that .

9) .

For instance, , , and . Indeed, arises in string theory trying to renormalize the vacuum energy of an infinite number of harmonic oscillators. The result in the bosonic string is . In order to match with Riemann zeta function regularization of the above series, the bosonic string is asked to live in an ambient spacetime of D=26 dimensions. We also have that

10) . The Riemann zeta value at the infinity is equal to the unit.

11) The derivative of the zeta function is . Particularly important of this derivative are:

or

This allow us to define the factorial of the infinity as

and the renormalized infinite dimensional determinant of certain operator A as:

, with

12) . This is a result used by theoretical physicists in dimensional renormalization/regularization. is the so-called Euler-Mascheroni constant.

This can be thought as “bosons made of fermions” or “fermions made of bosons” somehow. Special values of Dirichlet eta function are given by:

Remark(I): is important in the physics realm, since the spectrum of the hydrogen atom has the following aspect

and the Balmer formula is, as every physicist knows

Remark (II): The fact that is finite implies that the energy level separation of the hydrogen atom in the Böhr level tends to zero AND that the sum of ALL the possible energy levels in the hydrogen atom is finite since is finite.

Remark(III): What about an “atom”/system with spectrum ? If , we do know that is the case of the Kepler problem. Moreover, it is easy to observe that corresponds to tha harmonic oscillator, i.e., . We also know that is the infinite potential well. So the question is, what about a spectrum and so on?

In summary, does the following spectrum

with energy separation/splitting

exist in Nature for some physical system beyond the infinite potential well, the harmonic oscillator or the hydrogen atom, where , and respectively?

It is amazing how Riemann zeta function gets involved with a common origin of such a different systems and spectra like the Kepler problem, the harmonic oscillator and the infinite potential well!

The Riemann Hypothesis (RH) is the greatest unsolved problem in pure Mathematics, and likely, in Physics too. It is the statement that the only non-trivial zeroes of the Riemann zeta function, beyond the trivial zeroes at have real part equal to 1/2. In other words, the equation or feynmanity has only the next solutions:

I generally prefer the following projective-like version of the RH (PRH):

The Riemann zeta function can be sketched on the whole complex plane, in order to obtain a radiography about the RH and what it means. The mathematicians have studied the critical strip with ingenious tools an frameworks. The now terminated ZetaGrid project proved that there are billions of zeroes IN the critical line. No counterexample has been found of a non-trivial zeta zero outside the critical line (and there are some arguments that make it very unlikely). The RH says that primes “have music/order/pattern” in their interior, but nobody has managed to prove the RH. The next picture shows you what the RH “say” graphically:

If you want to know how the Riemann zeroes sound, M. Watkins has done a nice audio file to see their music.

Riemann zeroes are connected with prime numbers through a complicated formula called “the explicit formula”. The next equation holds integer numbers, and non-trivial Riemann zeroes in the complex (upper) half-plane with :

and where is the celebrated Gauss prime number counting function, i.e., represents the prime numbers that are equal than x or below. This explicit formula was proved by Hadamard. The explicit formula follows from both product representations of , the Euler product on one side and the Hadamard product on the other side.

The function , sometimes written as , is the logarithmic integral

The explicit formula comes in some cool variants too. For instance, we can write

Date: January 3, 1982. Andrew Odlyzko wrote a letter to George Pólya about the physical ground/basis of the Riemann Hypothesis and the conjecture associated to Polya himself and David Hilbert. Polya answered and told Odlyzko that while he was in Göttingen around 1912 to 1914 he was asked by Edmund Landau for a physical reason that the Riemann Hypothesis should be true, and suggested that this would be the case if the imaginary parts, say of the non-trivial zeros

of the Riemann zeta function corresponded to eigenvalues of an unbounded and unknown self adjoint operator . That statement was never published formally, but it was remembered after all, and it was transmitted from one generation to another. At the time of Pólya’s conversation with Landau, there was little basis for such speculation. However, Selberg, in the early 1950s, proved a duality between the length spectrum of a Riemann surface and the eigenvalues of its Laplacian. This so-called Selberg trace formula shared a striking resemblance to the explicit formula of certain L-function, which gave credibility to the speculation of Hilbert and Pólya.

Dialogue(circa 1970). “(…)Dyson: So tell me, Montgomery, what have you been up to? Montgomery: Well, lately I’ve been looking into the distribution of the zeros of the Riemann zeta function. Dyson: Yes? And? Montgomery: It seems the two-point correlations go as….(…) Dyson: Extraordinary! Do you realize that’s the pair-correlation function for the eigenvalues of a random Hermitian matrix? It’s also a model of the energy levels in a heavy nucleus, say U-238.(…)”

A step further was given in the 1970s, by the mathematician Hugh Montgomery. He investigated and found that the statistical distribution of the zeros on the critical line has a certain property, now called Montgomery’s pair correlation conjecture. The Riemann zeros tend not to cluster too closely together, but to repel. During a visit to the Institute for Advanced Study (IAS) in 1972, he showed this result to Freeman Dyson, one of the founders of the theory of random matrices. Dyson realized that the statistical distribution found by Montgomery appeared to be the same as the pair correlation distribution for the eigenvalues of a random and “very big/large” Hermitian matrix with size NxN. These distributions are of importance in physics and mathematics. Why? It is simple. The eigenstates of a Hamiltonian, for example the energy levels of an atomic nucleus, satisfy such statistics. Subsequent work has strongly borne out the connection between the distribution of the zeros of the Riemann zeta function and the eigenvalues of a random Hermitian matrix drawn from the theoyr of the so-calle Gaussian unitary ensemble, and both are now believed to obey the same statistics. Thus the conjecture of Pólya and Hilbert now has a more solid fundamental link to QM, though it has not yet led to a proof of the Riemann hypothesis. The pair-correlation function of the zeros is given by the function:

In a posterior development that has given substantive force to this approach to the Riemann hypothesis through functional analysis and operator theory, the mathematician Alain Connes has formulated a “trace formula” using his non-commutative geometry framework that is actually equivalent to certain generalized Riemann hypothesis. This fact has therefore strengthened the analogy with the Selberg trace formula to the point where it gives precise statements. However, the mysterious operator believed to provide the Riemann zeta zeroes remain hidden yet. Even worst, we don’t even know on which space the Riemann operator is acting on.

However, some trials to guess the Riemann operator has been given from a semiclassical physical environtment as well. Michael Berry and Jon Keating have speculated that the Hamiltonian/Riemann operator is actually some kind of quantization of the classical Hamiltonian where is the canonical momentum associated with the position operator . If that Berry-Keating conjecture is true. The simplest Hermitian operator corresponding to is

At current time, it is still quite inconcrete, as it is not clear on which space this operator should act in order to get the correct dynamics, nor how to regularize it in order to get the expected logarithmic corrections. Berry and Germán Sierra, the latter in collaboration with P.K.Townsed, have conjectured that since this operator is invariant under dilatations perhaps the boundary condition for integer may help to get the correct asymptotic results valid for big . That it, in the large we should obtain

Indeed, the Berry-Keating conjecture opened another striking attack to prove the RH. A topic that was popular in the 80’s and 90’s in the 20th century. The weird subject of “quantum chaos”. Quantum chaos is the subject devoted to the study of quantum systems corresponding to classically chaotic systems. The Berry-Keating conjecture shed light further into the Riemann dynamics, sketching some of the properties of the dynamical system behind the Riemann Hypothesis.

In summary, the dynamics of the Riemann operator should provide:

1st. The quantum hamiltonian operator behind the Riemann zeroes, in addition to the classical counterpart, the classical hamiltonian , has a dynamics containing the scaling symmetry. As a consequence, the trajectories are the same at all energy scale.
2nd. The classical system corresponding to the Riemann dynamics is chaotic and unstable.
3rd. The dynamics lacks time-reversal symmetry.
4th. The dynamics is quasi one-dimensional.

A full dictionary translating the whole correspondence between the chaotic system corresponding to the Riemann zeta function and its main features is presented in the next table:

In 2001, the following paper emerged, http://arxiv.org/abs/nlin/0101014. The Riemannium arxiv paper was published later (here: Reg. Chaot. Dyn. 6 (2001) 205-210). After that, Brian Hayes wrote a really beautiful, wonderful and short paper titled The Spectrum of Riemannium in 2003 (American Scientist, Volume 91, Number 4 July–August, 2003,pages 296–300). I remember myself reading the manuscript and being totally surprised. I was shocked during several weeks. I decided that I would try to understand that stuff better and better, and, maybe, make some contribution to it. The Spectrum of Riemannium was an amazing name, an incredible concept. So, I have been studying related stuff during all these years. And I have my own suspitions about what the riemannium and the zeta function are, but this is not a good place to explain all of them!

The riemannium is the mysterious physical system behind the RH. Its spectrum, the spectrum of riemannium, are given by the RH and its generalizations.

Moreover, the following sketch from Hayes’ paper is also very illustrative:

Riemann zeta function also arises in the renormalization of the Standard Model and the regularization of determinants with “infinite size” (i.e., determinants of differential operators and/or pseudodifferential operators). For instance, the -dimensional regularized determinant is defined through the Riemann zeta function as follows:

The dimensional renormalization/regularization of the SM makes use of the Riemann zeta function as well. It is ubiquitous in that approach, but, as far as I know, nobody has asked why is that issue important, as I have suspected from long time ago.

Riemann zeta function is also used in the theory of Quantum Statistics. Quantum Statistics are important in Cosmology and Condensed Matter, so it is really striking that Riemann zeta values are related to phenomena like Bose-Einstein condensation or the Cosmic Microwave Background and also the yet to be found Cosmic Neutrino Background!

Let me begin with the easiest quantum (indeed classical) statistics, the Maxwell-Boltzmann (MB) statistics. In 3 spatial dimensions (3d) the MB distribution arises ( we will work with units in which ):

Usually, there are 3 thermodynamical quantities that physicists wish to compute with statistical distributions: 1) the number density of particles , 2) the energy density and 3) the pressure . In the case of a MB distribution, we have the following definitions:

We can introduce the dimensionless variables $late z=\dfrac{mc^2}{k_BT}$, . In this way,

With these definitions, the particle density becomes

This integral can be calculated in closed form with the aid of modified Bessel functions of the 2th kind:

or equivalently

And thus, we have the next results (setting for simplicity):

Even entropy density is easiy to compute:

These results can be simplified in some limit cases. For instance, in the massless limit . Moreover, we also know that . In such a case, we obtain:

We note that in this massless limit.

Remark (I): In the massless limit, and whenever there is no degeneracy, holds.

Remark (II): If there is a quantum degeneracy in the energy levels, i.e., if , we must include an extra factor of for massive particles of spin j. For massless photons with helicity, there is a degeneracy.

Remark (III): In the D-dimensional (D=d+1) Bose gas with dispersion relationship , it can be shown that the pressure is related with the energy density in the following way

Remark (IV): Let us define as the number of ways an integer number can be expressed as a sum of the sth powers of integers. For instance,

because

because

If with and , then and the partition function is

We will see later that

with is nothing but the generatin function of the partitions

The Hardy-Ramanujan inversion formula reads (for the case s=1 only):

Remark (V): There are some useful integrals in quantum statistics. They are the so-called Bose-Einstein/Fermi-Dirac integrals

The BE-FD quantum distributions in 3d are defined as follows:

where the minus sign corresponds to FD and the plus sign to BE.

We will firstly study the BE distribution in 3d. We have:

Introducing a scaled temperature , we get

Again, we can study a particularly simple case: the massless limit with . In this case, we get:

The FD distribution in 3d can be studied in a similar way. Following the same approach as the BE distribution, we deduce that:

and again the massless limit and provide

Remark (I): For photons with degeneracy we obtain

Remark (II): In Cosmology, Astrophysics and also in High Energy Physics, the following units are used

The Cosmic Microwave Background is the relic photon radiation of the Big Bang, and thus it has a temperature due to photons in the microwave band of the electromagnetic spectrum. Its value is:

Indeed, it also implies that the relic photon density is about

It is also speculated that there has to be a Cosmic Neutrino Background relic from the Big Bang. From theoretical Cosmology, it is related to the photon CMB temperature in the following way:

or equivalently

This temperature implies a relic neutrino density (per species, i.e., with ) about

The cosmological density entropy due to these particles is

and then we get

Remark (III): In Cosmology, for fermions in 3d ( note that BE implies , and that we must drop the factors in the next numerical values) we can compute

Remark (IV): An example of the computation of degeneracy factor is the quark-gluon plasma degeneracy . Firstly we compute the gluon and quark degeneracies

Then, the QG plasma degeneracy factor is

In general, for charged leptons and nucleons , for neutrinos (per species, of course), and for gluons and photons. Remember that massive particles with spin j will have .

Remark (V): For the Planck distribution, we also get the known result for the thermal distribution of the blackbody radiation

where . Moreover, there is certain isomorphism between the shift operator space and the space of functions through the map .

We define the generalized logarithm as the image under the previous map of . That is:

where , with , and . Furthermore, the next contraints are also given for every generalized logarithm:

1st. .

2nd. , , and .

3rd. , and where .

With these definitions we also have that

A)

B)

Examples of generalized logarithms are:

1) The Tsallis logarithm.

2) The Kaniadakis logarithm.

3) The Abe logarithm.

4) The biparametric logarithm.

with and in the case of the Abe logarithm.

Group entropies are defined through the use of generalized logarithms. Define some discrete probability distribution with normalization . Therefore, the group entropy is the following functional sum:

where we have used the previous definition of generalized logarithm and the Boltzmann’s constant is a real number. It is called group entropy due to the fact that is connected to some universal formal group. This formal group will determine some correlations for the class of physical systems under study and its invariant properties. In fact, the Tsallis logarithm itself is related to the Riemann zeta function through a beautiful equation! Under the Tsallis group exponential, the isomorphism is defined to be , and thus we easily get:

The primon gas/free Riemann gas is a statistical mechanics toy model illustrating in a simple way some correspondences between number theory and concepts in statistical physics, quantum mechanics, quantum field theory and dynamical systems.

The primon gas IS a quantum field theory (QFT) of a set of non-interacting particles, called the “primons”. It is also named a gas or a free model because the particles are non-interacting. There is no potential. The idea of the primon gas was independently discovered by Donald Spector (D. Spector, Supersymmetry and the Möbius Inversion Function, Communications in Mathemtical Physics 127 (1990) pp. 239-252) and Bernard Julia (Bernard L. Julia, Statistical theory of numbers, in Number Theory and Physics, eds. J. M. Luck, P. Moussa, and M. Waldschmidt, Springer Proceedings in Physics, Vol. 47, Springer-Verlag, Berlin, 1990, pp. 276-293). There have been later works by Bakas and Bowick (I. Bakas and M.J. Bowick, Curiosities of Arithmetic Gases, J. Math. Phys. 32 (1991) p. 1881) and Spector (D. Spector, Duality, Partial Supersymmetry, and Arithmetic Number Theory, J. Math. Phys. 39 (1998) pp.1919-1927) in which it was explored the connection of such systems to string theory.

2nd. The eigenenergies or spectrum are given by and they have energies proportional to . Mathematically speaking,

with

Please, note the natural emergence of a “free” scale of energy . What is this scale of energy? We do not know!

3rd. The second quantization/second-quantized version of this Hamiltonian converts states into particles, the “primons”. Multi-particle states are defined in terms of the numbers of primons in the single-particle states :

This corresponds to the factorization of into primes:

The labelling by the integer “N” is unique, since every number has a unique factorization into primes.

with , and where is the Boltzmann’s constant and T is the absolute temperature. The divergence of the zeta function at the value (corresponding to the harmonic sum) is due to the divergence of the partition function at certain temperature, usually called Hagedorn temperature. The Hagedorn temperature is defined by:

This temperature represents a limit beyond the system of (bosonic) primons can not be heated up. To understand why, we can calculate the energy

A similar treatment can be built up for fermions rather than bosons, but here the Pauli exclusion principle has to be taken into account, i.e. two primons cannot occupy the same single particle state. Therefore can be 0 or 1 for all single particle state. As a consequence, the many-body states are labeled not by the natural numbers, but by the square-free numbers. These numbers are sieved from the natural numbers by the Möbius function. The calculation is a bit more complex, but the partition function for a non-interacting fermion primon gas reduces to the relatively simple form

The canonical ensemble is of course not the only ensemble used in statistical physics. Julia extended the Riemann gas approach to the grand canonical ensemble by introducing a chemical potential (Julia, B. L., 1994, Physica A 203(3-4), 425), and thus, he replaced the primes p with new primes . This generalisation of the Riemann gas is called the Beurling gas, after the Swedish mathematician Beurling who had generalised the notion of prime numbers. Examining a boson primon gas with fugacity , it shows that its partition function becomes

Remarkable interpretation: pick a system, formed by two sub-systems not interacting with each other, the overall partition function is simply the product of the individual partition functions of the subsystems. From the previous equation of the free fermionic riemann gas we get exactly this structure, and so there are two decoupled systems. Firstly, a fermionic “ghost” Riemann gas at zero chemical potential and, secondly, a boson Riemann gas with energy-levels given by . Julia also calculated the appropriate Hagedorn temperatures and analysed how the partition functions of two different number theoretical gases, the Riemann gas and the “log-gas” behave around the Hagedorn temperature. Although the divergence of the partition function hints the breakdown of the canonical ensemble, Julia also claims that the continuation across or around this critical temperature can help understand certain phase transitions in string theory or in the study of quark confinement. The Riemann gas, as a mathematically tractable model, has been followed with much attention because the asymptotic density of states grows exponentially, , just as in string theory. Moreover, using arithmetic functions it is not extremely hard to define a transition between bosons and fermions by introducing an extra parameter, called kappa , which defines an imaginary particle, the non-interacting parafermions of order . This order parameter counts how many parafermions can occupy the same state, i.e. the occupation number of any state falls into the interval , and therefore belongs to normal fermions, while are the usual bosons. Furthermore, the partition function of a free, non-interacting κ-parafermion gas can be defined to be (Bakas and Bowick,1991, in the paper Bakas, I., and M. J. Bowick, 1991, J. Math. Phys. 32(7), 1881):

Indeed, Bakas et al. proved, using the Dirichlet convolution , how one can introduce free mixing of parafermions with different orders which do not interact with each other

where the symbol means d is a divisor of n. This operation preserves the multiplicative property of the classically defined partition functions, i.e., . It is even more intriguing how interaction can be incorporated into the mixing by modifying the Dirichlet convolution with a kernel function or twisting factor

Using the unitary convolution Bakas establishes a pedagogically illuminating case, the mixing of two identical boson Riemann gases. He shows that

This result has an amazing meaning. Two identical boson Riemann gases interacting with each other through the unitary twisting, are equivalent to mixing a fermion Riemann gas with a boson Riemann gas which do not interact with each other. Therefore, one of the original boson components suffers a transmutation/mutation into a fermion gas!

Remark (I): the Möbius function, which is the identity function with respect to the operation (i.e. free mixing), reappears in supersymmetric quantum field theories as a possible representation of the operator, where F is the fermion number operator! In this context, the fact that for square-free numbers is the manifestation of the Pauli exclusion principle itself! In any QFT with fermions, is a unitary, hermitian, involutive operator where is the fermion number operator and is equal to the sum of the lepton number plus the baryon number, i.e., , for all particles in the Standard Model and some (most of) SUSY QFT. The action of this operator is to multiply bosonic states by 1 and fermionic states by -1. This is always a global internal symmetry of any QFT with fermions and corresponds to a rotation by an angle . This splits the Hilbert space into two superselection sectors. Bosonic operators commute with whereas fermionic operators anticommute with it. This operator really is, therefore, more useful in supersymmetric field theories.

Remark (II): potential attacks on the Riemann Hypothesis may lead to advances in physics and/or mathematics, i.e., progress in Physmatics!

Remark (III): the energy of the ground state is taken to be zero and the energy spectrum of the excited state is , where , , runs over the prime numbers. Let N and E denote now the number of particles in the ground state and the total energy of the system, respectively. The fundamental theorem of arithmetic allows only one excited state configuration for a given energy

where n is an integer. It immediately means that this gas preserves its quantum nature at any temperature, since only one quantum state is permitted to be occupied. The number fluctuation of any state (even the ground state) is therefore zero. In contrast, the changes in the number of particles in the ground state predicted by the canonical ensemble is a smooth non-vanishing function of the temperature, while the grand-canonical ensemble still exhibits a divergence. This discrepancy between the microcanonical (combinatorial) and the other two ensembles remains even in the thermodynamic limit.

One could argue that the Riemann gas is fictitious/unreal and its spectrum is unrealisable/unphysical. However, we, physicists, think otherwise, since the spectrum does not increase with N more rapidly than , therefore the existence of a quantum mechanical potential supporting this spectrum is possible (e.g., via inverse scattering transform or supplementary tools). And of course the question is: what kind of system has such an spectrum?

Some temptative ideas for the potential based on elementary Quantum Mechanics will be given in the next section.

Instead of considering the free Riemann gas, we could ask to Quantum Mechanics if there is some potential providing the logarithmic spectrum of the previous section. Indeed, there exists such a potential. Let us factorize any natural number in terms of its prime “atoms”:

Take the logarithm

where are prime numbers (note that if we include “1” as a prime number it gives a zero contribution to the sum).

Now, suppose a logarithmic oscillator spectrum, i.e.,

with

with . In order to have a “riemann gas”/riemannium, we impose an spectrum labelled in the following fashion

Equivalently, we could also define the spectrum of interacting riemannium gas as

Massive elementary particles (with mass m) can be understood as composite particles made of confined particles moving with some energy inside a sphere of radius R. We note that we do not define futher properties of the constituent particles (e.g., if they are rotating strings, particles, extended objects like branes, or some other exotic structure moving in circular orbits or any other pattern as trajectory inside the composite particle).

Let us make the hypothesis that there is some force needed to counteract the centrifugal force . The centrifugal force is equal to , i.e., the balancing force F is . Then, assuming the two forces are equal in magnitude, we get

where is some constant, and that equation holds regardless the origin of the interaction. The potentail energy necessary to confine a constituent particle will be, in that case,

with some integration constant to be determined later. The center of mass of the “elementary particle”, truly a composite particle, from the external observer and the mass assinged to the composited system is:

The logarithmic potential energy is postulated to be proportional to , and it provides

with is another constant. In fact, are parameters that don’t depend, a priori, on the radius R but on the constitutent particle properties and coupling constants, respectively. Indeed, for instance, we could set and fix the ratio to the constant , where is the gravitational constant. However, such a constraint is not required from first principles or from a clear physical reason. From the following equations:

and

we get

Quantum Mechanics implies that the angular momentum should be quantized, so we can make the following generalization

so

Using the previous integral and this last result, we obtain

This is due to the fact that and

Combining these equations, we deduce the value of as a function of the parameters

The ratio can be calculated from the above equations as well, since

for the case n=0 implies that

, and after exponentiation, it yields

Introducing the variable we have to solve the equation

The solution is from which the relationship between and can be easily obtained. Indeed, we can make more deductions from this result. From , then

If we take , with , then

so

with and

Equivalently, the masses would be dynamically generated from the above equations, since

and

so we would deduce a particle spectrum given by a logarithmic spiral, through the equation

Remark: The shift implies that the spiral would begin with as the lowest mass and not the biggest mass, turning the spiral from inside to the outside region and vice versa.

In summary, the logarithmic oscillator is also related to some kind of confined particles and it provides a toy model of confinement!

Is the link between classical statistical mechanics and Riemann zeta function unique or is it something more general? C. Tsallis explained long ago the connection of non-extensive Tsallis entropies an the Riemann zeta function, given supplementary arguments to support the idea of a physical link between Physics, Statistical Mechanics and the Riemann hypothesis. His idea is the following.

A) Consider the harmonic oscillator with spectrum

, are the H.O. eigenenergies.

B) Consider theTsallis partition function

where and the deformed q-exponential is defined as

and

and the inverse of the deformed exponential is the q-logarithm

It implies that

Now, defining the Hurwitz zeta function as:

the last equation can be rewritten in a simple and elegant way:

This system can be called the Tsallis gas or the Tsallisium. It is a q-deformed version (non-extensive) of the free Riemann gas. And it is related to the harmonic oscillator! The issue, of course, is the problematic limit .

In the limit we get the Riemann zeta function from the Hurwitz zeta function:

or

The above equation, the partition function of the Tsallis gas/Tsallisium, connects directly the Riemann zeta function with Physics and non-extensive Statistical Mechanics. Indeed, C.Tsallis himself dedicated a nice slide with this theme to M.Berry:

For readers not familiarized with Tsallis generalized entropies, I would like to expose you the main definitions of such a generalization of classical statistical entropy (Boltzmann-Gibbs-Shannon), in a nutshell! I have to discuss more about this kind of statistical mechanics in the future, but today, I will only anticipate some bits of it.

Tsallis entropy (and its Statistical Mechanics/Thermodynamics) is based on the following entropy functionals:

1st. Discrete case.

plus the normalization condition

2nd. Continuous case.

plus the normalization condition

3rd. Quantum case. Tsallis matrix density.

plus the normatlization condition

In all the three cases above, we have defined the q-logarithm as , , and the 3 Tsallis entropies satisfy the non-additive property:

Theoretical physicsts suspect that Physics of the spacetime at the Planck scale or beyond will change or will be meaningless. There, the spacetime notion we are familiarized to loose its meaning. Even more, we could find those changes in the fundamental structure of the Polyverse to occur a higher scales of length. Really, we don’t know yet where the spacetime “emerges” as an effective theory of something deeper, but it is a natural consequence from our current limited knowledge of fundamental physics. Indeed, it is thought that the experimental device making measurements and the experimenter can not be distinguished at Planck scale. At Planck scale, we can not know at this moment how the framework of cosmology and the Hilbert space tool of Quantum Mechanics could be obtained with some unified formalism. It is one of the challenges of Quantum Gravity.

Many people and scientists think that geometry and topology of sub-Planckian lengths should not have any relation with our current geometry or topology. We say and believe that geometry, topology, fields and the main features of macroscopic bodies “emerge” from the ultra-Planckian and “subquantum” realm. It is an analogue to the colours of the rainbow emerging from the atoms or how Thermodynamics emerge from Statistical Mechanics.

There are many proposed frameworks to go beyond the usual notions of space and time, but the p-adic analysis approach is a quite remarkable candidate, having several achievements in its favor.

Motivations for a p-adic and adelic approaches as the ultimate substructure of the microscopic world arise from:

1) Divergences of QFT are believed to be absent with such number structures. Renormalization can be found to be unnecessary.

2) In an adelic approach, where there is no prime with special status in p-adic analysis, it might be more natural and instructive to work with adeles instead a pure p-adic approach.

3) There are two paths for a p-adic/adelic QM/QFT theory. The first path considers particles in a p-adic potential well, and the goal is to find solutions with smoothly varying complex-valued wavefunctions. There, the solutions share certain kind of familiarity from ordinary life and ordinary QM. The second path allows particles in p-adic potential wells, and the goal is to find p-adic valued wavefunctions. In this case, the physical interpretation is harder. Yet the math often exhibits surprising features and properties, and some people are trying to explores those novel and striking aspects.

Ordinary real (or even complex as well) numbers are familiar to everyone. Ostroswski’s theorem states that there are essentially only two possible completions of the rational numbers ( “fractions” you do know very well). The two options depend on the metric we consider:

1) The real numbers. One completes the rationals by adding the limit of all Cauchy sequences to the set. Cauchy sequences are series of numbers whose elements can be arbitrarily close to each other as the sequence of numbers progresses. Mathematically speaking, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other. Real numbers satisfy the triangle inequality .

2) The p-adic numbers. The completions are different because of the two different ways of measuring distance. P-adic numbers satisfy an stronger version of the triangle inequality, called ultrametricity. For any p-adic number is shows

Spaces where the above enhanced triangle inequality/ultrametricity arises are called ultrametric spaces.

In summary, there exist two different types of algebraic number systems. There is no other posible norm beyond the real (absolute) norm or the p-adic norm. It is the power of Mathematics in action.

Then, a question follows inmediately. How can we unify such two different notions of norm, distance and type of numbers. After all, they behave in a very different way. Tryingo to answer this questions is how the concept adele emerges. The ring of adeles is a framework where we consider all those different patterns to happen at equal footing, in a same mathematical language. In fact, it is analogue to the way in which we unify space and time in relativistic theories!

Adele numbers are an array consisting of both real (complex) and p-adic numbers! That is,

where is a real number and the are p-adic numbers living in the p-adic field . Indeed, the infinity symbol is just a consequence of the fact that real numbers can be thought as “the prime at infinity”. Moreover, it is required that all but finitely many of the p-adic numbers lie in the entire p-adic set . The adele ring is therefore a restricted direct (cartesian) product. The idele group is defined as the essentially invertible elements of the adelic ring:

We can define the calculus over the adelic ring in a very similar way to the real or complex case. For instance, we define trigonometric functions, , logarithms and special functions like the Riemann zeta function. We can also perform integral transforms like the Mellin of the Fourier transformation over this ring. However, this ring has many interesting properties. For example, quadratic polynomials obey the Hasse local-global principle: a rational number is the solution of a quadratic polynomial equation if and only if it has a solution in and for all primes p. Furthermore, the real and p-adic norms are related to each other by the remarkable adelic product formula/identity:

and where is a nonzero rational number.

Beyond complex QM, where we can study the particle in a box or in a ring array of atoms, p-adic QM can be used to handle fractal potential wells as well. Indeed, the analogue Schrödinger equation can be solved and it has been useful, for instance, in the design of microchips and self-similar structures. It has been conjectured by Wu and Sprung, Hutchinson and van Zyl,here http://arXiv.org/abs/nlin/0304038v1 , that the potential constructed from the non-trivial Riemann zeroes and prime number sequences has fractal properties. They have suggested that for the Riemann zeroes and for the prime numbers. Therefore, p-adic numbers are an excellent method for constructing fractal potential wells.

By the other hand, following Feynman, we do know that path integrals for quantum particles/entities manifest fractal properties. Indeed we can use path integrals in the absence of a p-adic Schrödinger equation. Thus, defining the adelic version of Feynman’s path integral is a necessary a fundamental object for a general quantum theory beyond the common textbook version. However, we need to be very precise with certain details. In particular, we have to be careful with the definition of derivatives and differentials in order to do proper calculations. Indeed we can do it since both, the adelic and idelic rings have a well defined translation-invariant Haar measure

and

These measures provide a way to compute Feynman path integrals over adelic/idelic spaces. It turns out that Gaussian integrals satisfy a generalization of the adelic product formula introduced before, namely:

where is an additive character from the adeles to complex numbers given by the map:

and is the fractional part of in the ordinary p-adic expression for x. This can be thought of as a strong generalization of the homomorphism .Then, the adelic path integral, with input parameters in the adelic ring and generating complex-valued wavefunctions follows up:

The eigenvalue problem over the adelic ring is given by:

where U is the time-development operator, are adelic eigenfunctions, and is the adelic energy. Here the notation has been simplified by using the subscript , which stands for all primes including the prime at infinity. One notices the additive character which allows these to be complex-valued integrals. The path integral can be generalized to p-adic time as well, i.e., to paths with fractal behaviour!

How is this p-adic/adelic stuff connected to the Riemannium an the Riemann zeta function? It can be shown that ground state of adelic quantum harmonic oscillator is

where is 1 if is a p-adic integer and 0 otherwise. This result is strikingly similar to the ordinary complex-valued ground state. Applying the adelic Mellin transform, we can deduce that

where are, respectively, the gamma function and the Riemann zeta function. Due to the Tate formula, we get that

.

and from this the functional equation for the Riemann zeta function naturally emerges.

In conclusion: it is fascinating that such simple physical system as the (adelic) harmonic oscillator is related to so significant mathematical object as the Riemann zeta function.

The Veneziano amplitude is also related to the Riemann zeta function and string theory. A nice application of the previous adelic formalism involves the adelic product formula in a different way. In string theory, one computes crossing symmetric Veneziano amplitudes describing the scattering of four tachyons in the 26d open bosonic string. Indeed, the Veneziano amplitude can be written in terms of Riemann zeta function in this way:

These amplitudes are not easy to calculate. However, in 1987, an amazingly simple adelic product formula for this tachyonic scattering was found to be:

Using this formula, we can compute and calculate the four-point amplitudes/interacting vertices at the tree level exactly, as the inverse of the much simpler p-adic amplitudes. This discovery has generated a quite a bit of activity in string theory, somewhat unknown, although it is not very popular as far as I know. Moreover, the whole landscape of the p-adic/adelic framework is not as easy for the closed bosonic string as the open bosonic strings (note that in a p-adic world, there is no “closure” but “clopen” segments instead of naive closed intervals). It has also been a source of controversy what is the role of the p-adic/adelic stuff at the level of the string worldsheet. However, there is some reasearch along these lines at current time.

Another nice topic is the vacuum energy and its physical manifestations. There are some very interesting physical effects involving the vacuum energy in both classical and quantum physics. The most important effects are the Casimir effect (vacuum repulsion between “plates”) , the Schwinger effect ( particle creation in strong fields) , the Unruh effect ( thermal effects seen by an uniformly accelerated observer/frame) , the Hawking effect (particle creation by Black Holes, due to Black Hole Thermodynamcis in the corresponding gravitational/accelerated environtment) , and the cosmological constant effect (or vacuum energy expanding the Universe at increasing rate on large scales. Itself, does it gravitate?). Riemann zeta function and its generalizations do appear in these 4 effects. It is not a mere coincidence. It is telling us something deeper we can not understand yet. As an example of why zeta function matters in, e.g., the Casimir effect, let me say that zeta function regularizes the following general sum:

Remark: I do know that I should have likely said “the cosmological constant problem”. But as it should be solved in the future, we can see the cosmological constant we observe ( very, very smaller than our current QFT calculations say) as “an effect” or “anomaly” to be explained. We know that the cosmological constant drives the current positive acceleration of the Universe, but it is really tiny. What makes it so small? We don’ t know for sure.

Remark(II): What are the p-adic strings/branes? I. Arefeva, I. Volovich and B. Dravogich, between other physicists from Russia and Eastern Europe, have worked about non-local field theories and cosmologies using the Riemann zeta function as a model. It is a relatively unknown approach but it is remarkable, very interesting and uncommon. I have to tell you about these works but not here, not today. I went too far, far away in this log. I apologize…

I have explained why I chose The Spectrum of Riemannium as my blog name here and I used the (partial) answer to explain you some of the multiple connections and links of the Riemann zeta function (and its generalizations) with Mathematics and Physics. I am sure that solving the Riemann Hypothesis will require to answer the question of what is the vibrating system behind the spectral properties of Riemann zeroes. It is important for Physmatics! I would say more, it is capital to theoretical physics as well.

Let me review what and where are the main links of the Riemann zeta function and zeroes to Physmatics:

2) The Riemannium has spectral properties similar to those of Random Matrix Theory.

3) The Hilbert-Polya conjecture states that there is some mysterious hamiltonian providing the zeroes. The Berry-Keating conjecture states that the “quantum” hamiltonian corresponding to the Riemann hypothesis is the corresponding or dual hamiltonian to a (semi)classical hamiltonian providing a classically chaotic dynamics.

4) The logarithmic potential provides a realization of certain kind of spectrum asymptotically similar to that of the free Riemann gas. It is also related to the issue of confinement of “fundamental” constituents inside “elementary” particles.

5) The primon gas is the Riemann gas associated to the prime numbers in a (Quantum) Statistical Mechanics approach. There are bosonic, fermionic and parafermionic/parabosonic versions of the free Riemann gas and some other generalizations using the Beurling gas and other tools from number theory.

6) The non-extensive Statistical Mechanics studied by C. Tsallis (and other people) provides a link between the harmonic oscillator and the Riemann hypothesis as well. The Tsallisium is the physical system obtained when we study the harmonic oscillator with a non-extensive Tsallis approach.

7) An adelic approach to QM and the harmonic oscillator produces the Riemann’s zeta function functional equation via the Tate formula. The link with p-adic numbers and p-adic zeta functions reveals certain fractal patterns in the Riemann zeroes, the prime numbers and the theory behind it. The periodicity or quasiperiodicity also relates it with some kind of (quasi)crystal and maybe it could be used to explain some behaviour or the prime numbers, such as the one behind the Goldbach’s conjecture.

8) A link between entropy, information theory and Riemann zeta function is done through the use of the notion of group entropy. Connections between the Veneziano amplitudes, tachyons, p-adic numbers and string theory arise after the Veneziano amplitude in a natural way.

9) Riemann zeta function also is used in the regularization/definition of infinite determinants arising in the theory of differential operators and similar maps. Even the generalization of this framework is important in number theory through the uses of generalizations of the Riemann zeta function and other arithmetical functions similar to it. Riemann zeta function is, thus, one of the simplest examples of arithmetical functions.

10) There are further links of the Riemann zeta function and “vacuum effects” like the Schwinger effect ( pair creating in strong fields) or the Casimir effect ( repulsive/atractive forces between close objects with “nothing” between them). Riemann zeta function is also related to SUSY somehow, either by the striking similarity between the Dirichlet eta function used in Fermi-Dirac statistics or directly with the explicit relationship between the Möbius function and the operator appearing in supersymmetric field theories.

In summary, Riemann zeta function is ubiquitious and it appears alone or with its generalizations in very different fields: number theory, quantum physics, (semi)classical physics/dynamics, (quantum) chaos theory, information theory, QFT, string theory, statistical physics, fractals, quasicrystals, operator theory, renormalization and many other places. Is it an accident or is it telling us something more important? I think so. Zeta functions are fundamental objects for the future of Physmatics and the solution of Riemann Hypothesis, perhaps, would provide such a guide into the ultimate quest of both Physics and Mathematics (Physmatics) likely providing a complete and consistent description of the whole Polyverse.

Then, the main unanswered questions to be answered are yet:

A) What is the Riemann zeta function? What is the riemannium/tsallisium and what kind of physical system do they represent really? What is the physical system behind the Riemann non-trivial zeroes? What does it mean for the Riemann zeroes arising from the Riemann zeta function generalizations in form of L-functions?

B) What is the Riemann-Hilbert-Polya operator? What is the space over the Riemann operator is acting?

C) Are Riemann zeta function and its generalization everywhere as they seem to be inside the deepest structures of the microscopic/macroscopic entities of the Polyverse?

I suppose you will now understand better why I decided to name my blog as The Spectrum of Riemannium…And there are many other reasons I will not write you here since I could reveal my current research.

However, stay tuned!

Physmatics is out there and everywhere, like fractals, zeta functions and it is full of lots of wonderful mathematical structures and simple principles!

We are going to learn about the different notions of velocity that the special theory of relativity provides.

The special theory of relativity is a simple wonderful theory, but it comes with many misconceptions due to bad teaching/science divulgation. It is not easy to master the full theory of relativity without the proper mathematical background and physical insight. In the internet era where knowledge is shared, a fundamental issue is to understand things properly. There are many people who thinks they understand the theory of relativity when they don’t. Even at the academia.

Moreover, you can find many people in the blogsphere/websphere trying to sell false theories and wrong theories. It is the same like the so-called alternative medicine: they are not medicine at all. Bad science is not science, it is simply a lie and not science at all. It is religion. Science can be critized, but nobody can critize that Earth revolves around the Sun, it is common knowledge and truth. So, we can make critics to scientist, but not the scientific method and well established theories. We can try to understand better or in a novel way, but we can not deny facts and experiments. Gerard ‘t Hooft, Nobe Prize, explain it in his web page www.phys.uu.nl/~thooft/.

It is important to remark that Science revolutions come when we extend the theories we know they are correct, like special relativity and not with a full destruction of the current and well-tested theories. Newtonian relativity is a limit of General Relativity. Galilean relativity is a limit of Special Relativity. Quantum Mechanics is a limit of QFT and so on. The issue is not that. Said these words, I am quite sure that scientists and particularly physicists wish to overcome current theories with new ones. However, the process to create a new theory is not easy. Specially, if you don’t understand the traps and theories that have passed every known test till now.

What is velocity? Classically, the answer is short and very clear/neat: velocity is the rate of change of position with respect to time. It is a vector magnitude. Mathematically speaking is the quotient between the displacement vector and the time interval, or in the infinitesimal limit, the derivative of the position vector with respect to time.

In the special theory of relativity, due to the fact that time is not universal but relative we can build different notions of velocity. And it matters. There are some clear concepts from relativity you should master till now:

a) You can attach a clock to any yardstick you could physically use for measurements of space and time.

b) You must distinguish the notions of coordinate velocity (map coordinate is another commonly used notion/concept) and proper velocity. The latter is sometimes called hyperbolic (or imaginary) velocity. These two notions are caused by the presence of two “natural” elections of time: the proper time and the coordinate time.

c) Due to the previous two facts, you must also distinguish between proper acceleration and geometric acceleration. Proper-accelerations caused by the tug of external forces and geometric accelerations caused by choice of a reference frame that’s not geodesic i.e. a local reference coordinate-system that is not ”in free-fall”. Proper-accelerations are felt through their points of action e.g. through forces on the bottom of your feet. On the other hand geometric accelerations give rise to inertial forces that act on every ounce of an object’s being. They either vanish when seen from the vantage point of a local free-float frame, or give rise to non-local force effects on your mass distribution that cannot be made to disappear. Coordinate acceleration goes to zero whenever proper-acceleration is exactly canceled by that connection term, and thus when physical and inertial forces add to zero.

People who are not aware of the previous comments, don’t understand relativity and the physics behind it. They even don’t undertand what experiments and their data say.

Let me review the main magnitudes, 3-vectors and 4-vectors which the special theory of relativity studies in the next tables:

The two notions of 3-velocity we do have from the special theory of relativity, i.e., from the 4-velocity , are:

1) Coordinate velocity, :

It is the common notion of 3-velocity, measured from an inertial observer with respect to the coordinate time t. Note that the coordinate time is not a true invariant in SR!

where is the proper time. This velocity can intuitively defined as the distance per unit traveler-time, retains many of the properties that ordinary velocity loses at high speed. In addition to these two definitions, we also have:

1)Proper-acceleration , is the acceleration experienced relative to a locally co-moving free-float-frame, and it helps when we are accelerating, speeding, and in curvy space-time.

2) How some of the space-like effect of sideways ”felt” forces moves into the reference-frame’s time-domain at high speed, making the relatively unknown bound (from special relativity!)

With the above definitions, the relativistic momentum can be expressed in termns of coordinate velocity or proper velocity as follows:

where

is the Lorentz factor. The last equal sign in the previous equation can be easily derived from the relativistic relationship:

and the definition of above.

Thanks to the metric-equation’s assignment of a frame-invariant traveler or proper-time to the displacement between events in context of a single map-frame of comoving yardsticks and synchronized clocks, proper velocity becomes one of three related derivatives in special relativity (coordinate velocity , proper-velocity , and Lorentz factor ) that describe an object’s rate of travel. For unidirectional motion, in units of lightspeed c (i.e. c=1 if we want to) each of these is also simply related to a traveling object’s hyperbolic velocity angle or rapidity by the next set of equations:

The next table illustrates how the proper-velocity of or “one map-lightyear per traveler-year” is a natural benchmark for the transition from a sub-relativistic coordinate frame to a (fake) auxiliary super-relativistic motion (in imaginary units of ). Note that the velocity angle or pseudorapidity and the proper-velocity run from 0 to infinity and track the physical coordinate-velocity when . On the other hand when , the (hyperbolic or imaginary) proper-velocity tracks Lorentz factor while velocity angle is logarithmic and hence increases much more slowly:

LUDICROUS SPEED AND WARP SPEED

Hyperbolic velocities CAN exceed c! They can reach even the ludicrous speed of when the coordinate velocity approaches c! However, you must never forget the fact that the velocity-angle/hyperbolic velocity IS imaginary in value. It is quite clear from the above table. Indeed, being somehow “trekkie” or a Sci-Fi “romantic” person, you could “define” warp-speeds as “imaginary/hyperbolic” velocities, i.e., in terms of proper velocity. In that case, you could get the correspondence

In general, we can define the WARP speed as and so, the proper velocity can be expressed in terms of the warp speed W in a very simple way . Thus, the real or coordinate velocity would be connected with warp-speed through the relativistic equation:

Of course, the point is that, unlike the Sci-Fi franchise, the real velocity has never exceeded c, only the hyperbolic velocity and the proper velocity (note that in terms of SR, velocities approaching c imply very boosted frames, so despite we could travel to any point of the Universe in SR only approaching c very closely with respect to the traveler proper time-one human life-, but in terms of the “Earth” (or rest) reference frame millions of years would have passed away!).

When the coordinate-speeds approach c, the respective coordinate velocities deviate from this simple addition rule in that rapidities (hyperbolic velocity angle boosts) add instead of velocities, i.e. . Coordinate velocities add non-linearly. And it is a well-tested consequence of the Special Theory of relativity. For highly relativistic objects (i.e. those with momentum per unit mass much larger than lightspeed) the result of the coordinate-velocity expression familiar from most textbooks is rather uninteresting since the coordinate-velocities all peak out at c, i.e., as everybody knows, in special relativity , because applying the relativistic addition of velocities rule, we get

And it is a fact from both theory and experiment! It will remain as long as SR remains a valid theory. SR holds yet with an astonishing degree of precision and accuracy. So, you can not deny every data and experiment that confirms SR. That is completely nonsense but there are some people and pseudo-scientists out there building their own theories AGAINST the achievements and explanations that SR provides to every experiment we have done until the current time. I am sorry for all of them. They are totally wrong. Science is not what they say it is. Any theory going beyond SR HAS to explain every experiment and data that SR does explain, and it is not easy to build such a theory or to say, e.g., why we have not observed (apparently) superluminal objects. I will discuss more superluminal in a forthcoming post/log entry, some posts after the special 50th post/log that is coming after this one! Stay tuned!

Coming back to our discussion…Why is all this stuff important? High Energy Physics is the natural domain of SR! And there, SR has not provided ANY wrong result till, in spite that some researches going beyond the Standard Model include modified dispersion relationships that reduce to SR in the low energy regime, we have not seen yet ANY deviation from SR until now.

For unidirectional motion, at low speeds the coordinate velocity of object 1 from the point of view of oncoming object 3 might be described as the sum of the velocity of object 1 with respect to lab frame 2 plus the velocity of the lab frame 2 with respect to object 3, that is:

Compare this expression to the previously obtained expression for rapidities! Rapidities always add, coordinate velocities add (linearly) only at low velocities. In conclusion, you must be careful by what you mean by velocity is a boosted system!

By the other hand, for relative proper-velocity, the result is:

This expression shows how the momentum per unit mass as well as the map-distance traveled per unit traveler time of object 1, as seen in the frame of oncoming particle 3, goes as the sum of the coordinate-velocities times the product of the gamma (energy) factors. The proper velocity equation is especially important in high energy physics, because colliders enable one to explore proper-speed and energy ranges much higher than accessible with fixed-target collisions. For instance each of two electrons (traveling with frames 1 and 3) in a head-on collision traveling in the lab frame (2) at

or equivalenty lightseconds per traveler second would see the other coming toward them at coordinate velocity and lightseconds per traveler second or . From the target’s view, that is an incredible increase in both energy and momentum per unit of mass.

Other magnitudes and their frame dependence in SR can be read from the following table:

CAUTION: These results don’t mean that the “real” energy is that. Energy is relative and it depends on the frame! The fact that in colliders, seen from the target reference frame, the energy can be greater than the center of mass energy is not an accident. It is a consequence of the formalism of special relativity. A similar observation can be done for velocities. Coordinate velocities, IN THE FRAMEWORK OF SPECIAL RELATIVITY, can never exceed the speed of light. As long as SR holds, there is no particle whose COORDINATE velocity can overcome the speed of light. However, we have seen that PROPER velocities are other monsters. They serve as a tool to handle rotations along the temporal axis, i.e., to handle boosts mixing space and time coordinates. Proper (or hyperbolic) velocities CAN be greater than speed of light. But, it does not contradict the special theory of relativity at all since hyperbolic velocities ARE NOT REAL since they are imaginary quantities and they are not physical. We can only measure momentum and real quantities! Moreover, remember that, in fact, group or phase velocities we have found before can ALSO be greater than c. So, you must be careful by what do you mean by velocity in SR or in any theory. Furthermore, you must distinguish the notion of particle velocity with those of the relative velocity between two inertial frames, since the particle velocities ( coordinate or proper) always refer to some concrete frame! In summary, be aware of people saying that there are superluminal particles in our colliders or astrophysical processes. It is simply not true. Superluminal objects have observable consequences, and they have failed to be observed ( the last example was the superluminal neutrino affair by the OPERA collaboration, now in agreement with SR).

Remark (I): From the last table we observe that in SR, the rotation angle is imaginary. Therefore, we are forced to use this gadget of hyperbolic velocity in order to avoid “imaginary velocities”.

Remark (II): Hyperbolic velocities would become imaginary velocities if we used the imaginary formalism of SR, the infamous .

Remark (III): Hyperbolic velocities are not coordinate velocities, so they are not physical at all. They are just a tool to provide the right answers in terms of rapidities, or the hyperbolic angle, whose units are imaginary radians! Hyperbolic velocities are measured in imaginary units of velocity!

Remark (IV): About the imaginary issues you can have now. The spacetime separation formula means that the time t can often be treated mathematically as if it were an imaginary spatial dimension. That is, you can define so , where is the square root of -1, and is a “fourth spatial coordinate”. Of course it is not at all. It is only a trick to treat the problem in a clever way. By the other hand, a Lorentz boost by a velocity can likewise be treated as a rotation by an imaginary angle. Consider a normal spatial rotation in which a primed frame is rotated in the -plane clockwise by an angle about the origin, relative to the unprimed frame. The relation between the coordinates and of a point in the two frames is:

Now set and , with both real. In other words, take the spatial coordinate to be imaginary, and the rotation angle likewise to be imaginary. Then the rotation formula above becomes

This agrees with the usual Lorentz transformation formulat if the boost velocity and boost angle are related by the known formula . We realize that if we identify the imaginary angle with the rapidity, we are back to Special Relativity. Indeed, it is only the rotations involving the time axis which can cause confusion because they are so different from our everyday experience. That is, we experience rotations along some direction in our daily experience, so we are familiarized with rotations and their (real) rotation angles. However, rotations along a time axis mixing space and time is a weird creature. It uses imaginary numbers or, if we avoid them, we have to use hyperbolic (pseudo)-rotations.